M. GROMOV Institut des nautes ]~tudes Scientifiques SIGN AND GEOMETRIC MEANING OF CURVATURE (Conferenza tenuta il 14 giugno 1990) SUNTO. -- Questa monografia ~ una versione estesa di una mia Conferenza tenuta in Milano nell'ambito delle Lezioni Leonardesche. E' un tentativo di rivelare ai non iniziati il meccanismo di sviluppo della geometria Riemanniana seguendo le tracce di relativamente poehe idee dalle fondamenta al tetto del- l'edificio. CONTENTS. w 0. The second fundamental form and convexity in the Euclidean space . Pag. 10 w Generalized convexity ..... >> 18 w 1. Recollection on length, distance and Riemannian metric >> 29 w 2. Equidistant deformation and the sectional curvature K(V) >> 38 w 2 Influence of K(V) on small balls in V . >~ 46 w 3. Manifolds with positive sectional curvature. ~ 49 w 3 Distance function and Alexandrov-Toponogov theorem . >> 59 w 3~. Singular metric spaces with K _>_ 0 ....... >> 61 w 3%. The sphere theorem and equidistant deformation of immersed hypersurfaces ...... 63 w Negative sectional curvature 70 w Ricci curvature . ~ 75 w Positive scalar curvature ..... >> 87 w 6 Spinors and the Dirac operator . >> 95 w The curvature operator and related invariants . >~ 101 w 7 Harmonic maps of surfaces and the complexified curvature Kr 106 w 7~. Harmonic maps into manifolds with Kr ___ 0 >> 117 w 7%. Metric classes defined by infinitesimal convex cones >> 120 10 M. GROMOV The curvature tensor of a Riemannian manifold is a little monster of (multi)linear algebra whose full geometric meaning remains obscure. However, one can define using the curvature several significant classes of manifolds and then these can be studied in the spirit of the old-fashioned synthetic geometry with no appeal to the world of infinitesimals where curvature tensors reside. A similar interplay between infinitesimal quantities and visual features of geometric objects appears in all corners of geo- metry and analysis. The simplest example is provided by tht equivalence of the two definitions of a monotone function df > 0 ~ f(tl) < f(t2) for t~ < to dt ....... Then the infinitesimals of the second order bring along a geome- trically more interesting phenomenon of convexity. (1 ) 1 d~f > 0 ~ f - (tl + t:) < -2- (f(tl + f(t2)) dt ~ - _ . Our next example lies at the very verge of the Riemannian domain so we look at in a greater detail. w 0. THE SECOND FUNDAMENTAL FORM AND CONVEXITY IN THE EUCLIDEAN SPACE. The basic infinitesimal invariant of a smooth hypersurface Wc ~" (<<hyper>> means codim W = n--dimW=l) is the de] second fundamental form 17 = 17 W which is the field of quadratic forms//~ on the tangents spaces T~(W) c T~(~') = ~ defined as follows. AFFINE DEFINITION OF H. Move w to the origine of ~" by a parallel translation of W in ~" and compose the resulting embedding W c ~ with the linear quotient map ~'--> ~'/Tw(W) ---- Nw. Identify the a,j (one dimensional linear) space Nw with ~ and thus obtain a function, say p ~ pw : W--> ~, whose differential Dp vanishes at w (by the definition of T~(W)). Then define the second differential De p that is a quadratic form on T~(W), such that for every pair of vector fields 01 and 02 on W the (second Lie) derivative of p at w satisfies 01(02 p) (w) = (D: p) (01(w), (0~(w)). SIGN AND GEOMETRIC MEANING OF CURVATURE 11 (The existence of such De p follws from Dp = 0 by a simple com- putation). This construction applies to all w E W and gives us our// for//w = D e P, thought of as a quadratic form on T(W) with values in the normal bundle N=Tw(1R~)/T(W) over W, where Tw(1R~) denotes the restriction T(IR~)IW. Here is the familiar picture for n = 1. W N w "-R Fig. 1. Geometrically speaking, Hw measures the second order infini- tesimal deviation of W from the affine subspace Tw(W)c ~. In particular, if W is connected, then the vanishing of // on W is equivalent to W lying in a hyperplane, as everybody knows. A more interesting relation between H and the (affine) geometry of W reads The farm II is positive semi-definite if and only if W is convex. To make it precise we have to choose a coo~ientation of W that is a way to distinguish between two components into which W locally divides 1R~. This is customary done with a transversal (e.g. normal) vector field v along W. Such a preferred field, once chosen, is called 12 Yl. GROMOV interior looking and the part of ~" where v looks into it is called the interiar of W. w terior W r w Fig. 2. Notice that such a field also defines an orientation of the normal fibers N~ and so one can speak of the sign of the forms llw with values in N~. Now we invoke the following DEFINITION OF CONVEXITY. W is called convex at w if it is contained in the interior half-space T + c ~" bounded by the hyper- plane T~(W) ~ "IR". For example, the curve W in Fig. 2 is convex at w but not at w'. Yet it becomes locally convex at w' if we change the coorientation by inverting the sign of ~. In the language of the projection p : W--~ N~ ~ ~ the con- vexity claim is p >__ 0 which well agrees with the positivity of II~--Dep. In fact, positive definiteness of H~ obviously implies local convexity (i.e. the convexity of small neighbourhood U c W of w) of W. But it is slightly harder to derive local convexity of W from positive semidefiniteness of H in a neighbourhood of w (rather than at w alone). Also, the proof of the global convexity of W is not quite trivial. Recall that the global convexity, (i.a the convexity SIGN AND GEOMETRIC MEANING OF CURVATURE 13 at all w ~ W) follows from H >_ 0, provided W is a closed connected hypersurface, where <( closed )) means compact without boundary. The affine definition of//given above is quite general. Namely, it applies to all dimensions and codimensions (but we need codim W ~ 1 to speak of convexity), it makes sense for arbitrary smooth maps f: W--> 1Rm (not only embeddings) and it generalizes to non- Euclidean ambient spaces endowed with affine connections. However, the affine nature of this definition makes it poorly adapted to the needs of the Riemannian geometry where the major object of study is the distance function associated to the Riemannian structure. With this in mind we turn to our second definition of// which is based on the following important notion. EQUIDISTANT DEFORMATION. Let W be a cooriented hyporsurface in 1R" and denote by J(x), x~ 1R" the signed distance function to W. That is ~(x)-----dist(x, W) for the exterior points x and 5(x)~---dist(x, W) in the interior. Notice that in general the distinction between interior and exterior points makes sense only locally near W and then J(x) is defined only in some small neigh- bourhood of W. Also recall that dist(x, W) ~- infdist(x, W) def wE ~r for the Euclidean distance dist(x, w)~ l]x--wll ~ < x --w, x --w > 1/2. Then we look at the levels of ~, that are w~ ~--~-l(e)~ {x ~ 1R" I ~(x)-- s}, and call them e-equidistant hypersurfaves or e-equidistaz~t defo~vna- tians of W ~ Wo. It is easy to show that for small e the manifolds W~ are smooth if Wo is smooth, but W~ may become singular for large c. In fact we shall see in a minute that the i~ward (i.e. e < 0) equidistant deformation necessarily produces singularities for every convex initial hypersurface Wo. (See fig. 3 below). For example such a deformation of the round sphere Wo~S~-l(r)c 1R" of radius r 14 M. GROMOV brings Wo to the center of the sphere for E-~r. (Here W = S'-l(r ~- ~) for all ~ >__ ~r). We r Wo interior N~~~~w Fig. 3. Next we consider the lines Nw in 1R" normal to W at the points w E W. It is easy to show that every such line meets each W~, for small e, at a single point denoted w, or (w, e) E W, and the resulting map d~ : W--> W~ for d~(w)= we is smooth. (In fact, d~ is a dif- feomorphism; moreover, Nw is normal to W, at w,, as elementary differential geometry tells us). Now we are going to define the second quadratic form as the rate of change of the lengths of curves C c Wo as we pass from Wo to infinitesimally close hypersurface W~. We recall that the length of C is determined by (integration of) the length of the tangent vectors of C which is given, in turn, by the first fu~tal form g on W that is just the restriction of the Euclidean scalar product (which is a quadratic form on ~) to the tangent spaces T~(W) c T~( IR ~) = 1R", w E W. In other words g is the Riemannian metric on W induced from the standard Riemannian metric on 1R~+1. Geometricly speaking, << induced ) amounts to the relation g-length(C) = The Euclidean length(C) for all smooth curves C in W. SIGN AND GEOMETRIC MEANING OF CURVATURE 15 We denote by g, the first fundamental form of W,, we pull it back to W by the differential of the map de and we denote the pulled back form g* on W ~ W~.